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Fundamentals of Statistical Signal Processing, Volume I: Estimation Theory Statistical and Adaptive Signal Processing - Spectral Estimation Signal Modeling. Fundamentals of statistical signal processing: estimation theory .. Blind detection in symmetric non-Gaussian noise with unknown PDF using maximum entropy. Signal Processing: Estimation Theory xi. 1 Introduction. 1. Estimation in Signal Processing. 1 7B Asymptotic PDF of MLE for a Scalar Parameter.

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Fundamentals of Statistical Signal Processing: Estimation Theory. Acoustic Waves: . 15B Derivation of Properties of Complex Gaussian PDF. 15C Derivation of. - Ebook download as PDF File .pdf), Text File .txt) or view presentation slides online. Fundamentals of Statistical Signal Processing--Estimation Theory-Kay - Ebook download as PDF File .pdf), Text File .txt) or view presentation slides online.

Population genetics Abstract In the population genomics era, the study of Y-chromosome variability is still of the greatest interest for several fields ranging from molecular anthropology to forensics and genetic genealogy. To provide new valuable estimates and to address these issues, we typed 47 Y-STRs comprising Yfiler, PowerPlex23 and YfilerPlus loci, the recently defined Rapidly Mutating [RM] panel and 11 additional markers often used in genetic genealogical applications in individuals belonging to 66 deep-rooting paternal genealogies from Northern Italy. Our results confirmed that the genealogy approach is an effective way to obtain reliable Y-STR mutation rate estimates even with a limited number of samples. Moreover, they showed that the impact of multi-step mutations and backmutations is negligible within the temporal scale usually adopted by forensic and genetic genealogy analyses. We then detected a significant association between the number of mutations within genealogies and observed TMRCAs.

Stoica, R. Moses Spectral Analysis of Signals.

Prentice hall, Lectures on Wiener and Kalman Filtering. Springer-Verlag, Wien — New York, Kailath, A. Sayed, B. Hassibi Linear Estimation Prentice Hall, Notions from Optimization. Description Parameter estimation : Random parameters, Bayesian estimation : minimum mean squared error estimation, orthogonality principle, maximum a posteriori estimation, performance bounds, linear estimation, the linear model.

Deterministic unknown parameters : minimum variance estimation, bias, efficiency, consistency, Cramer-Rao lower bound, maximum likelihood estimation, EM algorithm, least-squares and BLUE methods, method of moments, the linear model. Spectrum estimation : Non-parametric techniques : periodogram, windowing, spectral leakage and resolution. Parametric techniques : autoregressive processes, linear prediction, maximum entropy, Levinson and Schur algortihms, lattice filters.

Time and frequency domain localization, short-time Fourier transform, wavelet transform, QMF, subbands, perfect reconstruction filter banks. Optimal filtering : Wiener filtering, non-causal, causal and FIR, application to channel equalization. Kalman filtering : time-varying and time-invariant state-space models. Application to channel tracking.

However, this is not a general rule, since there are marked differences in patterns of mitochondrial DNA and Y-chromosome variation between matrilocal and patrilocal populations 8.

All these features, combined to the recent advances in sequencing technologies, guarantee to Y-chromosome studies an important role even in the genomics and post-genomics era.

Potential applications range from reconstructing male-mediated migration events in the past, Y-chromosomal variation from ancient DNA, the impact on genetic variability of male-related social-cultural features such as patrilineal systems of land inheritance, male-related social stratification, etc. The most important applications of Y-STRs in forensic investigation, such as paternity tests, individual identification and familial searching, rely on the possibility of tracing a relationship between two or more men and on the distinctiveness of individual haplotypes 3.

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Up to now Y-STR typing in forensics as well as in molecular anthropology mostly relied on a limited number of markers which were implemented in a few widely used kits, e. Further developments in the forensic field brought the identification of a set of 13 Rapidly Mutating RM Y-STRs that are characterized by particularly high mutation rates median: 1.

Of these, only a handful is available in the most complete commercial kits, namely PowerPlexY23 and YfilerPlus. Their high mutability makes RM Y-STRs ideal tools for increasing the differentiation of unrelated and especially related males, thus contributing to solve cases of homoplasy i. Besides this, RM Y-STRs have been shown to convey a significant phylogenetic signal, which may help to correctly classify individuals belonging to haplogroups characterized by strong demographic expansion and resemblance of Y-STR profiles such as the widespread Eurasian R-M lineage 26 , 27 , Different approaches were proposed by Zhivotovsky et al.

However, it has been suggested that the effects of homoplasy are negligible at the time-scale of forensic studies and genealogy-based research 28 , as well as for molecular anthropology studies aimed at reconstructing the most recent events of population history. The first aim of this study is to provide a wide set of Y-STR mutation rates in the genealogical time-frame for a set of Italian samples. Accordingly, we base our research on deep-rooting pedigrees from Italy and compare our results with those from previous studies that used a direct-count approach i.

To do this, we explore the relationship between genealogy depth based on archived records and the corresponding number of Y-STR mutations. As a methodological framework, we implemented the Walsh method 32 within an R script.

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Our dataset comprises individuals forming 63 paternally related pairs and three trios see Materials and Methods for details. Accordingly, the above mentioned seven pairs were excluded from downstream analyses, while the trio was reduced to a pair after removing the non-matching haplotype. This leads to a final set of 57 pairs and two trios for a total of individuals.

Calculation of mutation rates The total number of meioses encompassed in the 59 retained genealogies is , while the total number of observed mutations is , the vast majority of them , i. Potential multi-step cases are mostly represented by two-step mutations 14 , while only a handful of them shows higher numbers of steps 3 three-step and 2 four-step cases. Since Y-STR information for 43 additional paternal lineages sampled in the same geographic area of the present study Emilia-Romagna was available 9 , 28 , these data were included in calculations of mutation rates.

In fact, that way we could expand the dataset and increase the accuracy of our estimates, at the same time avoiding possible biases related to the geographic origins of the samples which could imply significantly different haplogroups composition and average generation times, both of them potentially influencing mutation rates PowerPlexY23 loci not included in the Yfiler include additional information from 14 paternal genealogies 28 corresponding to 92 added generations for a total of In this study, multi-copy Y-STRs were considered as single loci, calculating their mutation rates as the total number of observed mutations in all STR copies divided by the number of meioses.

Since all other pedigrees allowed straightforward counts, we decided to keep these loci and calculated mutation rates after having excluded the two ambiguous cases. As expected, six of them including the four most rapid ones are multi-copy STRs. Consistently with previous studies 28 , 33 , the relatively slow mutation rates of the loci included limit the power of Yfiler to completely resolve individual differentiation among related males.

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Full size image Mutation rates in meioses bins In order to check if mutation rates are influenced by changes in temporal depth of genealogies, we organized our data according to pairwise comparisons within genealogies , including previously published data 9 , 28 when overlapping with the Y-STR sets used in this study. In particular, mutation rates were calculated for three bin groups, based on the number of meioses separating pairs of individuals from their common ancestor, i.

Obtained results Fig. In fact, the corresponding confidence intervals are highly overlapping, thus suggesting no significant traces of saturation in the considered time frame i.

Full size image Average generation time Within the considered genealogies, we detected reliable information on the age of the fathers at the time of birth of their sons for meioses and the corresponding total number of years is 23, This leads to an estimate of average generation time equal to Relationship between the number of meioses and the number of observed mutations In theory, we would expect that the higher the number of meioses, the higher the number of observed mutations. Among them, the best-fitting ones are L, which includes the highest number of markers, and RM, which exhibits the highest mutation rates.

Analyses were limited to the whole dataset All and to the Y-STR panels that showed a significant and strong association between the number of observed mutations and the number of generations, i.

Digital Signal Processing - Journal - Elsevier

RM and L. As expected, an even higher fit was observed for the complete dataset All. In addition, All-based estimates yield tighter CIs combining the best match with the highest precision. By contrast, the least effective match was observed with the Leuven set and the SMM model This may be a consequence of the fact that the strict SMM does not take into account the possibility of multi-step mutation events, while considering changes in allele states as the increase or decrease by a single repeat unit at a time.

Therefore, the corresponding estimates tend to be higher than the IAM ones.

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Therefore, we suggest that the above procedure may be used as reliable tool for identifying non-paternity events with Y-STR data. We asked ourselves which of these three summary statistics best match with the observed values. As a first overview, we calculated the deviation between estimated and observed values by weighting them for the corresponding degrees of freedom. We notice that, while almost all summary statistics tend to overestimate the actual values, in general IAM estimates exhibit lower deviation than SMM ones.

As expected, deviations follow the same trend of the considered summary statistics, with the highest values for means and the lowest for modes.

Estimation theory

Therefore, modes of the posterior distributions seem to best approximate the observed values. S2 , Supplementary Information. Table 4 Deviation and parameters of both normal regression models and regressions through the origin, with observed TMRCA values as a function of the expected ones. Full size table Discussion Y-chromosomal studies still play a relevant role in current human genomics research.

Fundamentals of Statistical Signal Processing--Estimation Theory

Moreover, they are of great interest in interdisciplinary research, next to a world-wide audience of citizen science practitioners 4. Mutation rates are key parameters for understanding how Y-STRs impact the variability of human Y-chromosome and, consequently, for their use in forensic sciences, genetic genealogy, human population genetics and molecular anthropology.

For instance, knowledge of mutation rates is of the greatest importance for designing and predicting the discriminatory power of a given Y-STR panel